3.385 \(\int \frac{\cos (c+d x)}{a+b \sin ^3(c+d x)} \, dx\)

Optimal. Leaf size=144 \[ -\frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \sin (c+d x)+b^{2/3} \sin ^2(c+d x)\right )}{6 a^{2/3} \sqrt [3]{b} d}+\frac{\log \left (\sqrt [3]{a}+\sqrt [3]{b} \sin (c+d x)\right )}{3 a^{2/3} \sqrt [3]{b} d}-\frac{\tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} \sin (c+d x)}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} a^{2/3} \sqrt [3]{b} d} \]

[Out]

-(ArcTan[(a^(1/3) - 2*b^(1/3)*Sin[c + d*x])/(Sqrt[3]*a^(1/3))]/(Sqrt[3]*a^(2/3)*b^(1/3)*d)) + Log[a^(1/3) + b^
(1/3)*Sin[c + d*x]]/(3*a^(2/3)*b^(1/3)*d) - Log[a^(2/3) - a^(1/3)*b^(1/3)*Sin[c + d*x] + b^(2/3)*Sin[c + d*x]^
2]/(6*a^(2/3)*b^(1/3)*d)

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Rubi [A]  time = 0.0999976, antiderivative size = 144, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {3223, 200, 31, 634, 617, 204, 628} \[ -\frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \sin (c+d x)+b^{2/3} \sin ^2(c+d x)\right )}{6 a^{2/3} \sqrt [3]{b} d}+\frac{\log \left (\sqrt [3]{a}+\sqrt [3]{b} \sin (c+d x)\right )}{3 a^{2/3} \sqrt [3]{b} d}-\frac{\tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} \sin (c+d x)}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} a^{2/3} \sqrt [3]{b} d} \]

Antiderivative was successfully verified.

[In]

Int[Cos[c + d*x]/(a + b*Sin[c + d*x]^3),x]

[Out]

-(ArcTan[(a^(1/3) - 2*b^(1/3)*Sin[c + d*x])/(Sqrt[3]*a^(1/3))]/(Sqrt[3]*a^(2/3)*b^(1/3)*d)) + Log[a^(1/3) + b^
(1/3)*Sin[c + d*x]]/(3*a^(2/3)*b^(1/3)*d) - Log[a^(2/3) - a^(1/3)*b^(1/3)*Sin[c + d*x] + b^(2/3)*Sin[c + d*x]^
2]/(6*a^(2/3)*b^(1/3)*d)

Rule 3223

Int[cos[(e_.) + (f_.)*(x_)]^(m_.)*((a_) + (b_.)*((c_.)*sin[(e_.) + (f_.)*(x_)])^(n_))^(p_.), x_Symbol] :> With
[{ff = FreeFactors[Sin[e + f*x], x]}, Dist[ff/f, Subst[Int[(1 - ff^2*x^2)^((m - 1)/2)*(a + b*(c*ff*x)^n)^p, x]
, x, Sin[e + f*x]/ff], x]] /; FreeQ[{a, b, c, e, f, n, p}, x] && IntegerQ[(m - 1)/2] && (EqQ[n, 4] || GtQ[m, 0
] || IGtQ[p, 0] || IntegersQ[m, p])

Rule 200

Int[((a_) + (b_.)*(x_)^3)^(-1), x_Symbol] :> Dist[1/(3*Rt[a, 3]^2), Int[1/(Rt[a, 3] + Rt[b, 3]*x), x], x] + Di
st[1/(3*Rt[a, 3]^2), Int[(2*Rt[a, 3] - Rt[b, 3]*x)/(Rt[a, 3]^2 - Rt[a, 3]*Rt[b, 3]*x + Rt[b, 3]^2*x^2), x], x]
/; FreeQ[{a, b}, x]

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 634

Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Dist[(2*c*d - b*e)/(2*c), Int[1/(a +
b*x + c*x^2), x], x] + Dist[e/(2*c), Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] &
& NeQ[2*c*d - b*e, 0] && NeQ[b^2 - 4*a*c, 0] &&  !NiceSqrtQ[b^2 - 4*a*c]

Rule 617

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*Simplify[(a*c)/b^2]}, Dist[-2/b, Sub
st[Int[1/(q - x^2), x], x, 1 + (2*c*x)/b], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 204

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTan[(Rt[-b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[-b, 2]), x] /
; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 628

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(d*Log[RemoveContent[a + b*x +
c*x^2, x]])/b, x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rubi steps

\begin{align*} \int \frac{\cos (c+d x)}{a+b \sin ^3(c+d x)} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{1}{a+b x^3} \, dx,x,\sin (c+d x)\right )}{d}\\ &=\frac{\operatorname{Subst}\left (\int \frac{1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx,x,\sin (c+d x)\right )}{3 a^{2/3} d}+\frac{\operatorname{Subst}\left (\int \frac{2 \sqrt [3]{a}-\sqrt [3]{b} x}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx,x,\sin (c+d x)\right )}{3 a^{2/3} d}\\ &=\frac{\log \left (\sqrt [3]{a}+\sqrt [3]{b} \sin (c+d x)\right )}{3 a^{2/3} \sqrt [3]{b} d}+\frac{\operatorname{Subst}\left (\int \frac{1}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx,x,\sin (c+d x)\right )}{2 \sqrt [3]{a} d}-\frac{\operatorname{Subst}\left (\int \frac{-\sqrt [3]{a} \sqrt [3]{b}+2 b^{2/3} x}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx,x,\sin (c+d x)\right )}{6 a^{2/3} \sqrt [3]{b} d}\\ &=\frac{\log \left (\sqrt [3]{a}+\sqrt [3]{b} \sin (c+d x)\right )}{3 a^{2/3} \sqrt [3]{b} d}-\frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \sin (c+d x)+b^{2/3} \sin ^2(c+d x)\right )}{6 a^{2/3} \sqrt [3]{b} d}+\frac{\operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,1-\frac{2 \sqrt [3]{b} \sin (c+d x)}{\sqrt [3]{a}}\right )}{a^{2/3} \sqrt [3]{b} d}\\ &=-\frac{\tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} \sin (c+d x)}{\sqrt [3]{a}}}{\sqrt{3}}\right )}{\sqrt{3} a^{2/3} \sqrt [3]{b} d}+\frac{\log \left (\sqrt [3]{a}+\sqrt [3]{b} \sin (c+d x)\right )}{3 a^{2/3} \sqrt [3]{b} d}-\frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \sin (c+d x)+b^{2/3} \sin ^2(c+d x)\right )}{6 a^{2/3} \sqrt [3]{b} d}\\ \end{align*}

Mathematica [A]  time = 0.0560145, size = 116, normalized size = 0.81 \[ -\frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \sin (c+d x)+b^{2/3} \sin ^2(c+d x)\right )-2 \log \left (\sqrt [3]{a}+\sqrt [3]{b} \sin (c+d x)\right )+2 \sqrt{3} \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} \sin (c+d x)}{\sqrt{3} \sqrt [3]{a}}\right )}{6 a^{2/3} \sqrt [3]{b} d} \]

Antiderivative was successfully verified.

[In]

Integrate[Cos[c + d*x]/(a + b*Sin[c + d*x]^3),x]

[Out]

-(2*Sqrt[3]*ArcTan[(a^(1/3) - 2*b^(1/3)*Sin[c + d*x])/(Sqrt[3]*a^(1/3))] - 2*Log[a^(1/3) + b^(1/3)*Sin[c + d*x
]] + Log[a^(2/3) - a^(1/3)*b^(1/3)*Sin[c + d*x] + b^(2/3)*Sin[c + d*x]^2])/(6*a^(2/3)*b^(1/3)*d)

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Maple [A]  time = 0.047, size = 120, normalized size = 0.8 \begin{align*}{\frac{1}{3\,bd}\ln \left ( \sin \left ( dx+c \right ) +\sqrt [3]{{\frac{a}{b}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}-{\frac{1}{6\,bd}\ln \left ( \left ( \sin \left ( dx+c \right ) \right ) ^{2}-\sqrt [3]{{\frac{a}{b}}}\sin \left ( dx+c \right ) + \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}+{\frac{\sqrt{3}}{3\,bd}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{\sin \left ( dx+c \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-1 \right ) } \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(d*x+c)/(a+b*sin(d*x+c)^3),x)

[Out]

1/3/d/b/(a/b)^(2/3)*ln(sin(d*x+c)+(a/b)^(1/3))-1/6/d/b/(a/b)^(2/3)*ln(sin(d*x+c)^2-(a/b)^(1/3)*sin(d*x+c)+(a/b
)^(2/3))+1/3/d/b/(a/b)^(2/3)*3^(1/2)*arctan(1/3*3^(1/2)*(2/(a/b)^(1/3)*sin(d*x+c)-1))

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)/(a+b*sin(d*x+c)^3),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 2.8406, size = 1018, normalized size = 7.07 \begin{align*} \left [\frac{3 \, \sqrt{\frac{1}{3}} a b \sqrt{-\frac{\left (a^{2} b\right )^{\frac{1}{3}}}{b}} \log \left (-\frac{3 \, \left (a^{2} b\right )^{\frac{1}{3}} a \sin \left (d x + c\right ) + a^{2} + 3 \, \sqrt{\frac{1}{3}}{\left (2 \, a b \cos \left (d x + c\right )^{2} - 2 \, a b - \left (a^{2} b\right )^{\frac{2}{3}} \sin \left (d x + c\right ) + \left (a^{2} b\right )^{\frac{1}{3}} a\right )} \sqrt{-\frac{\left (a^{2} b\right )^{\frac{1}{3}}}{b}} + 2 \,{\left (a b \cos \left (d x + c\right )^{2} - a b\right )} \sin \left (d x + c\right )}{{\left (b \cos \left (d x + c\right )^{2} - b\right )} \sin \left (d x + c\right ) - a}\right ) - \left (a^{2} b\right )^{\frac{2}{3}} \log \left (-a b \cos \left (d x + c\right )^{2} + a b - \left (a^{2} b\right )^{\frac{2}{3}} \sin \left (d x + c\right ) + \left (a^{2} b\right )^{\frac{1}{3}} a\right ) + 2 \, \left (a^{2} b\right )^{\frac{2}{3}} \log \left (a b \sin \left (d x + c\right ) + \left (a^{2} b\right )^{\frac{2}{3}}\right )}{6 \, a^{2} b d}, \frac{6 \, \sqrt{\frac{1}{3}} a b \sqrt{\frac{\left (a^{2} b\right )^{\frac{1}{3}}}{b}} \arctan \left (\frac{\sqrt{\frac{1}{3}}{\left (2 \, \left (a^{2} b\right )^{\frac{2}{3}} \sin \left (d x + c\right ) - \left (a^{2} b\right )^{\frac{1}{3}} a\right )} \sqrt{\frac{\left (a^{2} b\right )^{\frac{1}{3}}}{b}}}{a^{2}}\right ) - \left (a^{2} b\right )^{\frac{2}{3}} \log \left (-a b \cos \left (d x + c\right )^{2} + a b - \left (a^{2} b\right )^{\frac{2}{3}} \sin \left (d x + c\right ) + \left (a^{2} b\right )^{\frac{1}{3}} a\right ) + 2 \, \left (a^{2} b\right )^{\frac{2}{3}} \log \left (a b \sin \left (d x + c\right ) + \left (a^{2} b\right )^{\frac{2}{3}}\right )}{6 \, a^{2} b d}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)/(a+b*sin(d*x+c)^3),x, algorithm="fricas")

[Out]

[1/6*(3*sqrt(1/3)*a*b*sqrt(-(a^2*b)^(1/3)/b)*log(-(3*(a^2*b)^(1/3)*a*sin(d*x + c) + a^2 + 3*sqrt(1/3)*(2*a*b*c
os(d*x + c)^2 - 2*a*b - (a^2*b)^(2/3)*sin(d*x + c) + (a^2*b)^(1/3)*a)*sqrt(-(a^2*b)^(1/3)/b) + 2*(a*b*cos(d*x
+ c)^2 - a*b)*sin(d*x + c))/((b*cos(d*x + c)^2 - b)*sin(d*x + c) - a)) - (a^2*b)^(2/3)*log(-a*b*cos(d*x + c)^2
+ a*b - (a^2*b)^(2/3)*sin(d*x + c) + (a^2*b)^(1/3)*a) + 2*(a^2*b)^(2/3)*log(a*b*sin(d*x + c) + (a^2*b)^(2/3))
)/(a^2*b*d), 1/6*(6*sqrt(1/3)*a*b*sqrt((a^2*b)^(1/3)/b)*arctan(sqrt(1/3)*(2*(a^2*b)^(2/3)*sin(d*x + c) - (a^2*
b)^(1/3)*a)*sqrt((a^2*b)^(1/3)/b)/a^2) - (a^2*b)^(2/3)*log(-a*b*cos(d*x + c)^2 + a*b - (a^2*b)^(2/3)*sin(d*x +
c) + (a^2*b)^(1/3)*a) + 2*(a^2*b)^(2/3)*log(a*b*sin(d*x + c) + (a^2*b)^(2/3)))/(a^2*b*d)]

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Sympy [A]  time = 15.9635, size = 260, normalized size = 1.81 \begin{align*} \begin{cases} \frac{\tilde{\infty } x \cos{\left (c \right )}}{\sin ^{3}{\left (c \right )}} & \text{for}\: a = 0 \wedge b = 0 \wedge d = 0 \\\frac{\sin{\left (c + d x \right )}}{a d} & \text{for}\: b = 0 \\- \frac{1}{2 b d \sin ^{2}{\left (c + d x \right )}} & \text{for}\: a = 0 \\\frac{x \cos{\left (c \right )}}{a + b \sin ^{3}{\left (c \right )}} & \text{for}\: d = 0 \\- \frac{\sqrt [3]{-1} \log{\left (- \sqrt [3]{-1} \sqrt [3]{a} \sqrt [3]{\frac{1}{b}} + \sin{\left (c + d x \right )} \right )}}{3 a^{\frac{2}{3}} b^{7} d \left (\frac{1}{b}\right )^{\frac{20}{3}}} + \frac{\sqrt [3]{-1} \log{\left (4 \left (-1\right )^{\frac{2}{3}} a^{\frac{2}{3}} \left (\frac{1}{b}\right )^{\frac{2}{3}} + 4 \sqrt [3]{-1} \sqrt [3]{a} \sqrt [3]{\frac{1}{b}} \sin{\left (c + d x \right )} + 4 \sin ^{2}{\left (c + d x \right )} \right )}}{6 a^{\frac{2}{3}} b^{7} d \left (\frac{1}{b}\right )^{\frac{20}{3}}} + \frac{\sqrt [3]{-1} \sqrt{3} \operatorname{atan}{\left (\frac{\sqrt{3}}{3} - \frac{2 \left (-1\right )^{\frac{2}{3}} \sqrt{3} \sin{\left (c + d x \right )}}{3 \sqrt [3]{a} \sqrt [3]{\frac{1}{b}}} \right )}}{3 a^{\frac{2}{3}} b^{7} d \left (\frac{1}{b}\right )^{\frac{20}{3}}} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)/(a+b*sin(d*x+c)**3),x)

[Out]

Piecewise((zoo*x*cos(c)/sin(c)**3, Eq(a, 0) & Eq(b, 0) & Eq(d, 0)), (sin(c + d*x)/(a*d), Eq(b, 0)), (-1/(2*b*d
*sin(c + d*x)**2), Eq(a, 0)), (x*cos(c)/(a + b*sin(c)**3), Eq(d, 0)), (-(-1)**(1/3)*log(-(-1)**(1/3)*a**(1/3)*
(1/b)**(1/3) + sin(c + d*x))/(3*a**(2/3)*b**7*d*(1/b)**(20/3)) + (-1)**(1/3)*log(4*(-1)**(2/3)*a**(2/3)*(1/b)*
*(2/3) + 4*(-1)**(1/3)*a**(1/3)*(1/b)**(1/3)*sin(c + d*x) + 4*sin(c + d*x)**2)/(6*a**(2/3)*b**7*d*(1/b)**(20/3
)) + (-1)**(1/3)*sqrt(3)*atan(sqrt(3)/3 - 2*(-1)**(2/3)*sqrt(3)*sin(c + d*x)/(3*a**(1/3)*(1/b)**(1/3)))/(3*a**
(2/3)*b**7*d*(1/b)**(20/3)), True))

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Giac [A]  time = 1.13095, size = 185, normalized size = 1.28 \begin{align*} -\frac{\frac{2 \, \left (-\frac{a}{b}\right )^{\frac{1}{3}} \log \left ({\left | -\left (-\frac{a}{b}\right )^{\frac{1}{3}} + \sin \left (d x + c\right ) \right |}\right )}{a} - \frac{2 \, \sqrt{3} \left (-a b^{2}\right )^{\frac{1}{3}} \arctan \left (\frac{\sqrt{3}{\left (\left (-\frac{a}{b}\right )^{\frac{1}{3}} + 2 \, \sin \left (d x + c\right )\right )}}{3 \, \left (-\frac{a}{b}\right )^{\frac{1}{3}}}\right )}{a b} - \frac{\left (-a b^{2}\right )^{\frac{1}{3}} \log \left (\sin \left (d x + c\right )^{2} + \left (-\frac{a}{b}\right )^{\frac{1}{3}} \sin \left (d x + c\right ) + \left (-\frac{a}{b}\right )^{\frac{2}{3}}\right )}{a b}}{6 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)/(a+b*sin(d*x+c)^3),x, algorithm="giac")

[Out]

-1/6*(2*(-a/b)^(1/3)*log(abs(-(-a/b)^(1/3) + sin(d*x + c)))/a - 2*sqrt(3)*(-a*b^2)^(1/3)*arctan(1/3*sqrt(3)*((
-a/b)^(1/3) + 2*sin(d*x + c))/(-a/b)^(1/3))/(a*b) - (-a*b^2)^(1/3)*log(sin(d*x + c)^2 + (-a/b)^(1/3)*sin(d*x +
c) + (-a/b)^(2/3))/(a*b))/d